5 Answers
5

A model that has shown some interest in recent years is the golden chain. In the golden chain you deal with a one-dimensional chain of spin-like particle, similar to the Heisenberg (or Ising) model. But in this model the spin degrees of freedom are replaced by (non-Abelian) anyons (see e.g. this thread). The type of anyon used in this model are the Fibonacci anyons.

To see how the golden ratio enters this model we have to look at the Hilbert space, specifically its dimension. In an ordinary spin chain each spin carries a degree of freedom to which we can assign a Hilbert space of dimension 2, $\mathcal{H}$. The tensor product of two spins is spanned by the singlet and triplet combination. The total Hilbert space of a chain with $n$ spins is the tensor product, $\otimes \mathcal{H}$ and it has dimension $2^n$.

Non-Abelian anyons on the other hand carry a different kind of spin. When two anyons combine they will form what is known as a fusion product. The fusion product of two anyons depends on the type of anyon you are dealing with. Fibonacci anyons satisfy the relation

$\tau \times \tau = 1 + \tau$

We can think of this analogous to spin (with a fundamental difference). When we bring two anyons together they will fuse together and form a composite-like particle. This is similar to the spin of two (s=1/2) particles which combine into a singlet (S=0) or doublet (S=1). In the case of the Fibonacci anyons the particle can form two type of composites: the vacuum particle $1$ ("zero spin") and the Fibonacci particle $\tau$.

What happens when we bring another $\tau$ \particle to this composite? It will fuse with the composite particle to form another composite. However, the allowed particles which can be formed depend on the fusion channel of the first two anyons:

There are two ways in which a $\tau$ particle is formed in the end, while there is only one way in which a vacuum particle is formed. All in all:

$\tau \times \tau \times \tau = 1+ 2\tau$

Where the factor of two on the right hand side refers to the number of ways in which a $\tau$ particle can be formed. The dimension of the Hilbert space of three tau particles is therefore 3 dimensional.

This gives the following conclusion:

We set the dimension of the Hilbert space of zero particles equal to 1.

The dimension of H for 1 Fib. anyon is also 1.

The dimension of H for 2 Fib. anyons is 2.

The dimension of H for 3 Fib. anyons is 3.

Any guess what the dimension of the 4 anyons will be? It's 5 dimensional. You can derive it yourself: just count the number of ways in which you can fuse the anyons together.

The sequence of the dimension of the Hilbert spaces for $n$ anyons is:

$1,1,2,3,5,8,13,\ldots$

Yes, this is the Fibonacci sequence! And the Fibonacci sequence has a very nice feature: It grows roughly as $\phi^n$ where $\phi$ is the golden ratio! The dimension of the Hilbert space of $n$ Fibonacci anyons grows roughly as $\phi^n$!

One way to think of this is that the Fibonacci anyons carry a spin of dimension $\phi$. This statement is wrong though: the Hilbert space always has an integer dimension. It is therefore not referred to as a spin but rather as the quantum dimension of the anyons. The rule is that on average the Hilbert space grows by a factor of $\phi$ every time you add an anyon to the chain (just like the H-space for an Ising chain grows with a factor of two every time you add a spin to the system).

On a last note: Fibonacci anyons might be realized in certain quantum Hall systems and they are useful for topological quantum computing, if they are ever found.

Actually the golden ratio does show up in at least one interesting physical situation: cobalt niobate is an experimental realization of the 1D Ising model, which -- in a magnetic field transverse to the axis that neighboring spins are coupled along -- has a quantum phase transition. In 1989 Zamoldchikov studied this model and discovered an amazing thing that has quite recently been verified experimentally by Coldea's group: if you look at magnetic excitations (through neutron scattering) right near the quantum critical point, there is structure at a sequence of energies -- the lowest two of these have a ratio that is exactly $\varphi$. [In fact it goes deeper than this, with the excitation energies actually having a connection to the structure of the E8 group... but we'll leave that as homework!]

How very interesting! Thanks for your response!
–
BKaylorMar 15 '11 at 4:13

3

Homework solution for anyone interested: Ising model at the critical point is described by conformal field theory. More precisely, it is a certain minimal model with central charge $c=1/2$ which can also be thought of as a representation of the affine Lie algebra based on $E_8$. But this isn't quite the model used in the experiment. Rather, the model is perturbed by additional transverse magnetic field and turns out that this leads to an affine $E_8$ Toda field theory. It is this latter theory that has 8 massive particles (with e.g. $m_2/m_1 = \phi$) as excitations.
–
MarekMar 15 '11 at 9:13

This is the most boring answer possible.
–
Keenan PepperMar 15 '11 at 1:38

1

Boring, but do you disagree? I guess we could try to sneak it in wherever we have sqrt(5).
–
nibotMar 15 '11 at 2:35

1

I disagree. It is used sometimes and it is also very often found in nature (think of Fibonacci numbers, logarithmic spirals, etc.). That you haven't seen it in some specialized field of yours doesn't really tells us anything. And so this answer is completely useless.
–
MarekMar 15 '11 at 9:18

2

@nibot, I woudn't call this a "toy" problem. This is the key to all tranmission line analysis, just insert complex resistances, and You have the model to phone lines, TV cables etc. The solution of such equation was the hard core of Pupins invention to revolutionize far dictance telephone cables (Pupin coils)
–
GeorgMar 15 '11 at 11:48

I googeled around for golden ratio yesterday, and when I saw the continued fraction formula for it, I spontaneously thought of tranmission lines. But I did not follow that thought. (I do not know how to treat continued fractions actually)
–
GeorgMar 15 '11 at 12:34

Googling arxiv comes up with lots of hits. For example:
NewJ.Phys.11:063026 (2009), Adisorn Adulpravitchai, Alexander Blum, Werner Rodejohann, Golden Ratio Prediction for Solar Neutrino Mixing :
It has recently been speculated that the solar neutrino mixing angle is connected to the golden ratio $\phi$. Two such proposals have been made, $\cot(\theta_{12}) = \phi$ and $\cos(\theta_{12}) = \phi/2$. We compare these Ansatze and discuss a model leading to $\cos(\theta_{12}) = \phi/2$ based on the dihedral group $D_{10}$. This symmetry is a natural candidate because the angle in the expression $\cos(\theta_{12} = \phi/2$ is simply $\pi/5$, or 36 degrees. This is the exterior angle of a decagon and $D_{10}$ is its rotational symmetry group. We also estimate radiative corrections to the golden ratio predictions.